Abstract

We examine the ranks of operators in semi-finite $\mathrm{C}^*$-algebras
as measured by their densely defined lower semicontinuous traces.
We first prove that a unital simple $\mathrm{C}^*$-algebra whose
extreme tracial boundary is nonempty and finite contains positive
operators of every possible rank, independent of the property
of strict comparison. We then turn to nonunital simple algebras
and establish criteria that imply that the Cuntz semigroup is
recovered functorially from the Murray-von Neumann semigroup
and the space of densely defined lower semicontinuous traces.
Finally, we prove that these criteria are satisfied by not-necessarily-unital
approximately subhomogeneous algebras of slow dimension growth.
Combined with results of the first-named author, this shows that
slow dimension growth coincides with $\mathcal Z$-stability,
for approximately subhomogeneous algebras.